Midterm I Past Exams UNKNOWN YEAR PDF

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MATH 251 - D100 & D300

1. Consider the line and the plane in R3 L : ~r (t) = h2, 0, 2i + th1, −1, 2i

P : x + y + z = 2.

[3]

(a) Find the point of intersection of the line L and the plane P .

[3]

(b) Find the vector equation of the line that passes through Q(0, 0, 1) and is perpendicular to the plane P .

[4]

(c) Find the scalar equation of the plane that contains both the point Q(0, 0, 1) and the line L.

MATH 251 - D100 & D300

2. For the vector function: √ ~r (t) = ( 2 sin t)iˆ + jˆ + (cos t) kˆ [4]

. Give the vector equation of the line, ~l (s), that is tangent to the space curve defined (a) by ~r (t) at t = π/4. Choose a parametrization for ~l (s) for which s = 0 corresponds to the point of tangency.

[2]

−→ −→ −→ (b) Let OP = ~r (π/3) and OQ = ~r (π), determine the vector QP and its magnitude.

[4]

(c) Find a vector function ~q(t) where

d~q (t) = ~r (t) and ~q (π) = h0, 1, 0i dt

MATH 251 - D100 & D300

3. Consider the function f (x, y) =

xy . + y2

2x2

[2]

(a) Over what domain is the function f continuous — explain.

[3]

(b) Show that the limiting values of f as (x, y) → (0, 0) are the same for paths that follow the x-axis and the y-axis.

[3]

(c) Give the limiting value of f as (x, y) → (0, 0) along a line with slope m in the xy-plane. What does this result imply?

[2]

4. Use concepts from Calculus I and/or limit theorems to explain why the following limit exists. lim

(x,y)→(0,0)

5xy 2 =0 x2 + y 2

MATH 251 - D100 & D300 [4]

5. Give a component-wise derivation of the vector identity: ˆ = − (jˆ · ~a) kˆ ˆj × (~a × k) and explain using orthogonality why the vector triple-product must be in the ˆk-direction.

6. Evaluate the following partial derivatives, and indicate what derivative rules are used. ∂z when z(x, y) = (4x2 y − 3xy 3 )10 ∂y

[3]

(a)

[3]

∂ (b) ∂x



xy x2 + y 2



MATH 251 - D100 & D300

7. Short answer questions, put your answers in the box. No partial credit (you do not need to justify your answer. [2]

(a) Compute the vector projection of ~v = h1, 1, 0i onto w ~ = h1, 1, 1i.

[2]

(b) Give the point on the line ~r (t) = h−1, 2, 3i + th−1, 1, 0i whose position vector is orthogonal to ~v = h1, 0, 1i.

[3]

(c) Determine the volume of the parallelepiped determined by the vectors v~1 = h1, 0, 4i, v~2 = h1, 3, 1i, v~3 = h−2, 1, 3i.

[3]

(d) True or False:

d [~r (t) × ~r ′ (t)] = −(~r ′′ (t) × ~r (t)). dt

MATH 251 - D100 & D300

8. A surface S is defined by the equation z(x, y ) = 3 − x2 − 4y 2 .

[1]

(a) Give the coordinates of the point P on the surface with x = 1 and y = 1.

[3]

(b) Consider the space curve obtained by intersecting the surface S with the plane y = 1. Determine the tangent vector to this curve at the point P .

[3]

(c) Identify and evaluate the partial derivative that gives the slope at P of the tangent to the space curve formed by the intersection of the surface S with the plane x = 1.

[3]

(d) Give a scalar equation of the tangent plane to the surface S at the point P ....